Subgroups — C₃₁×#22592 #21479351

Order 7936.

804 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1917	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1916	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1851	C₂	1	2	yes	maximal
5	C₃₁×#1839	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1817	C₂	1	1	yes	maximal, characteristic
8	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1753	C₂	1	2	yes	maximal
10	C₃₁×#1745	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
14	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
26	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#376	C₄	1	1	yes	characteristic
30	C₃₁×#376	C₄	1	1	yes	characteristic
31	C₄³×C₃₁	C₂²	1	1	yes	characteristic
32	C₄³×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#336	C₂²	1	1	yes	characteristic
34	C₃₁×#336	C₂²	1	1	yes	characteristic
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	1	yes	characteristic
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₃₁	—	2	4	no	—
47	C₂×M₁₆×C₃₁	—	2	4	no	—
48	C₂×M₁₆×C₃₁	—	4	2	no	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₃₁×#117	—	2	4	no	—
54	C₃₁×#117	—	2	4	no	—
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	—	2	4	no	—
64	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊C₈	—	2	4	no	—
81	C₃₁×C₄⋊C₈	—	4	2	no	—
82	C₃₁×C₂²⋊C₈	—	2	4	no	—
83	C₃₁×#97	—	2	4	no	—
84	C₄×C₈×C₃₁	—	4	2	no	—
85	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
86	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
87	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
88	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
89	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
90	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
91	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
92	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
93	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	4	1	no	—
103	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	#41	1	4	yes	—
117	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
118	M₁₆×C₃₁	—	4	8	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	#41	1	4	yes	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	complemented
135	C₂²⋊C₄×C₃₁	—	2	2	no	—
136	C₂²⋊C₄×C₃₁	—	2	2	no	—
137	C₂²⋊C₄×C₃₁	—	2	2	no	—
138	C₂²⋊C₄×C₃₁	—	2	2	no	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	2	2	no	—
143	C₄²×C₃₁	—	2	2	no	—
144	C₄²×C₃₁	—	2	2	no	—
145	C₄²×C₃₁	—	2	2	no	—
146	C₄²×C₃₁	—	2	2	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	2	1	no	—
152	C₄²×C₃₁	—	2	1	no	—
153	C₄²×C₃₁	—	2	1	no	—
154	C₄²×C₃₁	—	4	1	no	—
155	C₄²×C₃₁	—	4	1	no	—
156	C₄²×C₃₁	M₁₆	1	4	yes	complemented
157	#22592	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂³×C₃₁	—	2	1	no	—
159	C₂³×C₃₁	—	2	1	no	—
160	C₂³×C₃₁	—	2	1	no	—
161	C₂³×C₃₁	—	2	1	no	—
162	C₂³×C₃₁	—	2	1	no	—
163	C₂³×C₃₁	—	2	1	no	—
164	C₂³×C₃₁	#117	1	2	yes	—
165	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₃₁	—	2	8	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	4	no	—
184	C₂×C₄×C₃₁	—	2	4	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	4	4	no	—
201	C₂×C₄×C₃₁	#117	1	2	yes	—
202	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
203	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
204	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
205	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
206	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
207	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
208	C₂×C₄×C₃₁	#131	1	2	yes	—
209	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
210	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
211	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
213	C₈×C₃₁	—	2	4	no	—
214	C₈×C₃₁	—	4	2	no	—
215	C₈×C₃₁	—	4	2	no	—
216	C₈×C₃₁	—	4	2	no	—
217	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
218	#1917	C₂×C₃₁	1	2	yes	complemented
219	#1916	C₂×C₃₁	1	2	yes	complemented
220	#1851	C₂×C₃₁	1	2	yes	—
221	#1839	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#1817	C₂×C₃₁	1	1	yes	characteristic
224	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1753	C₂×C₃₁	1	2	yes	—
226	#1745	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#1728	C₂×C₃₁	1	1	yes	characteristic
228	C₂²×C₃₁	—	2	2	no	—
229	C₂²×C₃₁	—	2	2	no	—
230	C₂²×C₃₁	—	2	2	no	—
231	C₂²×C₃₁	—	4	1	no	—
232	C₂²×C₃₁	—	4	1	no	—
233	C₂²×C₃₁	—	4	1	no	—
234	C₂²×C₃₁	—	4	1	no	—
235	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
236	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
237	C₂²×C₃₁	#520	1	1	yes	characteristic
238	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
239	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
240	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
241	C₂²×C₃₁	#568	1	1	yes	characteristic
242	C₄×C₃₁	—	2	2	no	—
243	C₄×C₃₁	—	2	2	no	—
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	2	2	no	—
247	C₄×C₃₁	—	2	2	no	—
248	C₄×C₃₁	—	2	2	no	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	4	1	no	—
251	C₄×C₃₁	—	4	1	no	—
252	C₄×C₃₁	—	4	1	no	—
253	C₄×C₃₁	—	4	1	no	—
254	C₄×C₃₁	—	4	1	no	—
255	C₄×C₃₁	#551	1	2	yes	—
256	C₄×C₃₁	#552	1	2	yes	—
257	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
258	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
259	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
260	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
262	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
263	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
264	C₂×#97	C₂²×C₃₁	1	2	yes	—
265	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
266	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
267	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
268	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
269	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
270	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
271	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
272	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
273	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
274	#376	C₄×C₃₁	1	1	yes	characteristic
275	#376	C₄×C₃₁	1	1	yes	characteristic
276	C₄³	C₂²×C₃₁	1	1	yes	characteristic
277	C₄³	C₂²×C₃₁	1	1	yes	characteristic
278	#336	C₂²×C₃₁	1	1	yes	characteristic
279	#336	C₂²×C₃₁	1	1	yes	characteristic
280	#336	C₂²×C₃₁	1	2	yes	—
281	#336	C₂²×C₃₁	1	1	yes	characteristic
282	#336	C₂²×C₃₁	1	2	yes	—
283	#336	C₂²×C₃₁	1	2	yes	—
284	#336	C₂²×C₃₁	1	2	yes	—
285	#336	C₂²×C₃₁	1	2	yes	—
286	#336	C₂²×C₃₁	1	2	yes	—
287	#336	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×C₃₁	—	4	2	no	—
289	C₂×C₃₁	#2343	1	1	yes	characteristic
290	C₂×C₃₁	#2920	1	1	yes	characteristic
291	C₂×C₃₁	#2924	1	1	yes	characteristic
292	C₂×C₃₁	#2965	1	1	yes	characteristic
293	C₂×C₃₁	#2971	1	1	yes	characteristic
294	C₂×C₃₁	#2989	1	1	yes	characteristic
295	C₂×C₃₁	#2990	1	1	yes	characteristic
296	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
297	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
298	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×M₁₆	—	2	4	no	—
300	C₂×M₁₆	—	2	4	no	—
301	C₂×M₁₆	—	4	2	no	—
302	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
303	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
304	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
305	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
306	#117	—	2	4	no	—
307	#117	—	2	4	no	—
308	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
312	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
314	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
315	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
316	C₂×C₄²	—	2	4	no	—
317	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
318	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
319	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
320	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
321	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
322	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
325	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
326	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
327	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
328	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
329	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
330	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
333	C₄⋊C₈	—	2	4	no	—
334	C₄⋊C₈	—	4	2	no	—
335	C₂²⋊C₈	—	2	4	no	—
336	#97	—	2	4	no	—
337	C₄×C₈	—	4	2	no	—
338	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
339	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
340	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
341	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
342	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
343	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
344	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
345	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
346	C₃₁	#22592	1	1	yes	characteristic, direct product factor, Sylow
347	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	4	1	no	—
357	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
358	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
359	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
360	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
361	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
362	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
363	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
364	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
365	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
366	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
368	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
369	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	#41×C₃₁	1	4	yes	—
371	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
372	M₁₆	—	4	8	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	#41×C₃₁	1	4	yes	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	complemented
389	C₂²⋊C₄	—	2	2	no	—
390	C₂²⋊C₄	—	2	2	no	—
391	C₂²⋊C₄	—	2	2	no	—
392	C₂²⋊C₄	—	2	2	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	2	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	2	2	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	4	1	no	—
409	C₄²	—	4	1	no	—
410	C₄²	M₁₆×C₃₁	1	4	yes	complemented
411	C₂³	—	2	1	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	—	2	1	no	—
416	C₂³	—	2	1	no	—
417	C₂³	C₃₁×#117	1	2	yes	—
418	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄	—	2	8	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	C₃₁×#117	1	2	yes	—
455	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
456	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
457	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
458	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
459	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
460	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
461	C₂×C₄	C₃₁×#131	1	2	yes	—
462	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
463	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
464	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
465	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
466	C₈	—	2	4	no	—
467	C₈	—	4	2	no	—
468	C₈	—	4	2	no	—
469	C₈	—	4	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	—	4	1	no	—
477	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
478	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
479	C₂²	C₃₁×#520	1	1	yes	characteristic
480	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
481	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
482	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
483	C₂²	C₃₁×#568	1	1	yes	characteristic
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	2	2	no	—
487	C₄	—	2	2	no	—
488	C₄	—	2	2	no	—
489	C₄	—	2	2	no	—
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	4	1	no	—
493	C₄	—	4	1	no	—
494	C₄	—	4	1	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	1	no	—
497	C₄	C₃₁×#551	1	2	yes	—
498	C₄	C₃₁×#552	1	2	yes	—
499	C₂	—	4	2	no	—
500	C₂	C₃₁×#2343	1	1	yes	characteristic
501	C₂	C₃₁×#2920	1	1	yes	characteristic
502	C₂	C₃₁×#2924	1	1	yes	characteristic
503	C₂	C₃₁×#2965	1	1	yes	characteristic
504	C₂	C₃₁×#2971	1	1	yes	characteristic
505	C₂	C₃₁×#2989	1	1	yes	characteristic
506	C₂	C₃₁×#2990	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃₁×#1917	C₂	semidirect
3. C₃₁×#1916	C₂	semidirect
5. C₃₁×#1839	C₂	semidirect
6. C₃₁×#1831	C₂	semidirect
8. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
10. C₃₁×#1745	C₂	semidirect
134. C₄⋊C₄×C₃₁	M₁₆	semidirect
156. C₄²×C₃₁	M₁₆	semidirect
157. #22592	C₃₁	direct (N × H)
218. #1917	C₂×C₃₁	semidirect
219. #1916	C₂×C₃₁	semidirect
221. #1839	C₂×C₃₁	semidirect
222. #1831	C₂×C₃₁	semidirect
224. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
226. #1745	C₂×C₃₁	semidirect
346. C₃₁	#22592	direct (N × H)
388. C₄⋊C₄	M₁₆×C₃₁	semidirect
410. C₄²	M₁₆×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₃₁×#117	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1851	2
C₃₁×#1839	2
C₃₁×#1831	2
C₃₁×#1817	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1753	2
C₃₁×#1745	2
C₃₁×#1728	2
#22592	31

Subgroups — C31×#22592 #21479351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22592 #21479351